Problem 26 - Algebra and Trigonometry

Question

25–30 ? Factor the expression by grouping terms. $$ 3 x^{3}-x^{2}+6 x-2 $$

Step-by-Step Solution

Answer

The expression factors to \((3x - 1)(x^2 + 2)\).

Step 1: Group Terms

To factor by grouping, first divide the expression into two groups: \(3x^3 - x^2\) and \(6x - 2\). This makes it easier to identify common factors within each group.

Step 2: Factor Each Group

Identify and factor out the greatest common factor in each group. For the first group \(3x^3 - x^2\), factor out \(x^2\), giving \(x^2(3x - 1)\). In the second group \(6x - 2\), factor out the common factor \(2\), giving \(2(3x - 1)\).

Step 3: Combine Common Factors

Both groups now have a common binomial factor \((3x - 1)\). Factor this out, resulting in \((3x - 1)(x^2 + 2)\).

Step 4: Verify the Solution

Multiply the factors to ensure the original expression is obtained: \((3x - 1)(x^2 + 2) = 3x^3 - x^2 + 6x - 2\). The expression is correctly factored.

Key Concepts

Polynomial Factoring

Factoring polynomials is a method used to simplify expressions, solve equations, or find roots. It involves rewriting a polynomial as a product of its factors, which are simpler expressions.

• For example, the polynomial expression from our exercise, \(3x^3 - x^2 + 6x - 2\), can be made simpler through factoring.
• The goal is to express it as the product of simpler polynomials, which can help in solving algebraic equations or evaluating expressions more easily.

When you factor a polynomial, you're breaking it down into its building blocks. Think of it like taking apart a LEGO structure to see the individual pieces that make it up.

Common Factors

A key part of polynomial factoring is identifying the common factors within an expression. Common factors are terms that appear in all parts of the expression allowing them to be factored out.

• For instance, in our example, we first look for common factors in groups like \(3x^3 - x^2\) and \(6x - 2\).
• Identifying common factors, like \(x^2\) in the first group, simplifies expressions and helps in factoring further.

By pulling out common factors, you make the expression easier to handle, much like finding common ingredients in different recipes. This step is crucial before moving on to more complex factoring techniques.

Binomial Factors

Binomial factors are pairs of terms that can be multiplied together to give another expression. In this context, recognizing them helps to simplify the polynomial further.

• After recognizing and factoring out the greatest common factors in each group, you may find a repeated binomial factor.
• In the exercise, the expression \((3x - 1)\) serves as a common binomial factor for both groups.

Finding a binomial factor lets you condense several terms into a simplified product, much like simplifying a fraction by finding and eliminating common numerical factors.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. Factoring these expressions involves recognizing patterns and organizing them efficiently.

• The expression \(3x^3 - x^2 + 6x - 2\) is an algebraic expression consisting of terms separated by addition or subtraction.
• Breaking it into groups and factoring those groups helps in rewriting the expression in a more manageable form.

Understanding algebraic expressions and their structures allows for more effective manipulation and simplification, making it easier to perform operations or solve equations.